Problem Set 2
Physics 341
Due October 22
Some abbreviations: S  Shankar.
1
. The Dirac deltafunction has many nice properties. Show that
δ
(
ax
) =
δ
(
x
)

a

. Show that
the limit
δ
(
x
) = lim
a
→
0
1
π
a
x
2
+
a
2
satisfies the properties of the Dirac deltafunction. Namely, that
δ
(
x
) vanishes if
x
6
= 0 and
Z
+
∞
∞
δ
(
x
) = 1
.
2
. Let’s continue with an exercise involving GramSchmidt.
(i) Take the following three vectors in
R
3
,

e
1
i
= (1
,
1
,
1)
,

e
2
i
= (1
,
1
,
0)
,

e
3
i
= (1
,
0
,
1)
.
Construct an orthonormal basis using the GramSchmidt procedure starting with

e
1
i
.
(ii) Now let’s return to normalizable functions on the interval [

1
,
1]. Consider the inner
product
h
f

g
i
=
Z
1

1
f
*
(
x
)
g
(
x
)
dx
on this space. We can expand these functions in the basis
{
1
, x, x
2
, x
3
, . . .
}
but this is not an
orthonormal basis! We can build an orthonormal basis using GramSchmidt. For example,
we can choose
φ
0
=
1
√
2
,
φ
1
=
r
3
2
x,
for the first two basis elements. Please construct
φ
2
and
φ
3
. These orthogonal polynomials
are called Legendre polynomials.
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 Spring '10
 Sghy
 Physics, mechanics, Fourier Series, Hilbert space, inner product

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